Amplitude Versus Offset (AVO) inversion attempts to use an amplitude variation with offset from available surface seismic data to estimate a density as well as reflection coefficients of a P-wave velocity and/or a S-wave-velocity of an earth-model. AVO results are used to infer a lithology and a fluid property of a reservoir, which is an ultimate goal for hydrocarbon exploration. A fact is that AVO inversion directly associates subsurface rock properties with surface seismic data with a limitation imposed by amplitude variation with offset. Therefore, the AVO inversion is an effective technique and is widely used to study reservoir characterizations. For example, it can be used in oil, gas, coalbed methane, coal field mining and mineral explorations.
A basis of the AVO theory is derived from the Zoeppritz equation (1919) that describes relationships between an incident angle of a plane wave and reflection and transmission coefficients of compressional waves (P-waves) and shear waves (S-waves) at an impedance contrast interface (where there are velocity and density contrasts) in a perfectly welded media. Many studies have derived useful AVO equations for meeting practical requirements by approximating original Zoeppritz equations. Rueger (2002) and Downton et al. (2011) provided diverse forms of approximate AVO equations to predict fractured rock properties that are dependent on the effective anisotropic medium theory. In 2002, Chaisri provided some exact formulas (which are complicated) for reflection and transmission coefficients for non-welded contact boundary conditions, but did not provide any AVO approximate formula.
There is a viewpoint that the fractures are also reflection generators for reflecting waves propagating in the media (Schoenberg, 1980; Cui, lines and krebes, 2013). Consequently, recorded surface seismic data would inevitably contain fracture reflections when an acquisition reservoir contains fractures. This would cause an error in the prediction of rock properties of the host media when an inversion processing is applied with a conventional AVO equation. The reason is that the conventional AVO equation is based on an assumption of perfectly welded contact, but they considered the fractured medium as an effective anisotropic medium. In other words, the conventional AVO equation does not entirely treat the fracture reflections apart from the input seismic data when they attempt to invert rock properties of the host media for a fractured medium with impedance contrast.